Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ elements $x_1,\ldots,x_n$ are equal to each other. The $\mathcal{F}^n_1$ said quasitrivial and those $\mathcal{F}^n_n$ idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathc...

In this paper, we adopt minimal gravitational decoupling scheme to extend a non-static spherically symmetric isotropic composition anisotropic interior in $$f({\mathcal {R}},{\mathcal {T}},{\mathcal {R}}_{\textrm{ab}}{\mathcal {T}}^{\textrm{ab}})$$ theory. A geometric deformation is applied only on $$g_{rr}$$ metric component through which the modified field equations are separated into two set...

Frames which can be generated by the action of some operators (e.g. translation, dilation, modulation, ...) on a single element $f$ in a Hilbert space, called coherent frames. In this paper, we introduce a class of continuous frames in a Hilbert space $mathcal{H}$ which is indexed by some locally compact group $G$, equipped with its left Haar measure. These frames are obtained as the orbits of ...

Proving a conjecture of Talagrand, fractional version the ``expectation-threshold" Kalai and second author, we show that $p_c (\mathcal{F}) = O(q_f(\mathcal{F})\mathrm{log}\ \ell(\mathcal{F}))$ for any increasing family $\mathcal{F}$ on finite set $X$, where $p_c(\mathcal{F})$ $q_f(\mathcal{F})$ are threshold ``fractional expectation-threshold" $\mathcal{F}$, $\ell(\mathcal{F})$ is maximum size...

Let $n\ge2$ and $d_1,\ldots,d_n\ge2$ be integers, $\mathcal{F}$ a field. A vector $u\in\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ is called product if $u=u^{[1]}\otimes\cdots\otimes u^{[n]}$ for some $u^{[1]}\in\mathcal{F}^{d_1},\ldots,u^{[n]}\in\mathcal{F}^{d_n}$. basis composed of vectors basis. In this paper, we show that the maximum dimension subspaces $\mathcal{F}^{d_1}\otimes...

In this paper, we analyze the modified $f(\mathcal{R})$ gravity models in Friedmann--Lema\^{\i}tre--Robertson--Walker (FLRW) background. The actions of bouncing cosmology are studied under consideration different viable theory that can resolve difficulty singularity standard Big-Bang cosmology. Under theory, cosmological constraints plotted provisions cosmic-time, then investigated bounce circu...